Parenthesization/Examples/4

Example of Parenthesization

A word of $4$ elements can be parenthesized in $5$ distinct ways:

$\quad a_1 \paren {a_2 \paren {a_3 a_4} }$

$\quad a_1 \paren {\paren {a_2 a_3} a_4}$

$\quad \paren {a_1 a_2} \paren {a_3 a_4}$

$\quad \paren {a_1 \paren {a_2 a_3} } a_4$

$\quad \paren {\paren {a_1 a_2} a_3} a_4$

From Number of Distinct Parenthesizations on Word, the number of distinct parenthesizations of a word $w$ of $n$ elements is the Catalan number $C_{n - 1}$:

$C_{n - 1} = \dfrac 1 n \dbinom {2 \paren {n - 1} } {n - 1}$

For $n = 4$ we have:

$\ds C_4$

$=$

$\ds \dfrac 1 4 \dbinom {2 \times 3} 3$

$\ds $

$=$

$\ds \dfrac 1 4 \times \dfrac {6!} {3! \times 3!}$

Definition of Binomial Coefficient

$\ds $

$=$

$\ds \dfrac 1 4 \times \dfrac {6 \times 5 \times 4} {3 \times 2 \times 1}$

Definition of Factorial

$\ds $

$=$

$\ds 5$

$\blacksquare$

1971: George E. Andrews: Number Theory ... (previous) ... (next): $\text {3-4}$ Generating Functions
1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $42$
1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $42$